Descartes and the birth of analytic geometry (Source: ) Descartes' Geometry was an appendix to a larger work called Discourse on the Method of Properly Conducting One's Reason and of Seeking the Truth in the Sciences . In this work, Descartes set out to place human knowledge on a new, firm footing, an important task in an age beset with doubts and controversy and in which skepticism reigned. Descartes himself had seen service in warfare spurred by religious disagreements, and he was no stranger to scientific controversy, either. In 1633, hearing that Galileo had been condemned for teaching that the earth moved around the sun, Descartes abruptly decided not to publish a work defending the idea of a sun-centered univierse. Descartes found a model for proper reasoning in mathematics, especially in geometry, and his appendix on Geometry was meant to illustrate the effectiveness and usefulness of his method. His method was based on four basic rules for deducing knowledge in the manner of a geometry proof. His famous line, "I think, therefore I am," reveals the first firm piece of knowledge upon which his subsequent deductions were based.

In 1649, Descartes accepted an invitation from Queen Christine of Sweden to move to the Swedish court in Stokhlom and become her private tutor. His new employer, however, forced him to begin lessons at 5 a.m.! For a man who had stayed in bed till 11 since childhood, this early rising, combined with the cold climate, proved to be too much. He died of a lung inflammation on February 11, 1650. In his own words: I thought the following four [rules] would be enough, provided that I made a firm and constant resolution not to fail even once in the observance of them. The first was never to accept anything as true if I had not evident knowledge of its being so; that is, carefully to avoid precipitancy and prejudice, and to embrace in my judgment only what presented itself to my mind so clearly and distinctly that I had no occasion to doubt it. The second, to divide each problem I examined into as many parts as was feasible, and as was requisite for its better solution.

The third, to direct my thoughts in an orderly way; beginning with the simplest objects, those most apt to be known, and ascending little by little, in steps as it were, to the knowledge of the most complex; and establishing an order in thought even when the objects had no natural priority one to another. And the last, to make throughout such complete enumerations and such general surveys that I might be sure of leaving nothing out. These long chains of perfectly simple and easy reasonings by means of which geometers are accustomed to carry out their most difficult demonstrations had led me to fancy that everything that can fall under human knowledge forms a similar sequence; and that so long as we avoid accepting as true what is not so, and always preserve the right order of deduction of one thing from another, there can be nothing too remote to be reached in the end, or to well hidden to be discovered. Discours de la Méthode. 1637. Omnia apud me mathematica fiunt. With me everything turns into mathematics. Cogito ergo sum. "I think, therefore I am." Discours de la Méthode. 1637. It is not enough to have a good mind. The main thing is to use it well.

Discours de la Méthode. 1637. If you would be a real seeker after truth, you must at least once in your life doubt, as far as possible, all things. Discours de la Méthode. 1637. analytic geometry | Britannica.com Analytic geometry mathematics Analytic geometry, also called coordinate geometry, mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry. The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations . This correspondence makes it possible to reformulate problems in geometry as equivalent problems in algebra , and vice versa; the methods of either subject can then be used to solve problems in the other. For example, computers create animations for display in games and films by manipulating algebraic equations. Elementary analytic geometry Apollonius of Perga (c. 262–190 bc), known by his contemporaries as the “Great Geometer,” foreshadowed the development of analytic geometry by more than 1,800 years with his book Conics . He defined a conic as the intersection of a cone and a plane (see figure ).

Using Euclid’s results on similar triangles and on secants of circles, he found a relation satisfied by the distances from any point P of a conic to two perpendicular lines, the major axis of the conic and the tangent at an endpoint of the axis. These distances correspond to coordinates of P, and the relation between these coordinates corresponds to a quadratic equation of the conic. Apollonius used this relation to deduce fundamental properties of conics . See conic section . analysis (mathematics): Analytic geometry Further development of coordinate systems (see figure ) in mathematics emerged only after algebra had matured under Islamic and Indian mathematicians. (See mathematics: The Islamic world (8th–15th centuries) and mathematics, South Asian .) At the end of the 16th century, the French mathematician François Viète introduced the first systematic algebraic notation, using letters to represent known and unknown numerical quantities, and he developed powerful general methods for working with algebraic expressions and solving algebraic equations. With the power of algebraic notation, mathematicians were no longer completely dependent upon geometric figures and geometric intuition to solve problems.

The more daring began to leave behind the standard geometric way of thinking in which linear (first power) variables corresponded to lengths, squares (second power) to areas, and cubics (third power) to volumes, with higher powers lacking “physical” interpretation. Two Frenchmen, the mathematician-philosopher René Descartes and the lawyer-mathematician Pierre de Fermat , were among the first to take this daring step. Similar Topics Ceva’s theorem Descartes and Fermat independently founded analytic geometry in the 1630s by adapting Viète’s algebra to the study of geometric loci. They moved decisively beyond Viète by using letters to represent distances that are variable instead of fixed. Descartes used equations to study curves defined geometrically, and he stressed the need to consider general algebraic curves—graphs of polynomial equations in x and y of all degrees. He demonstrated his method on a classical problem: finding all points P such that the product of the distances from P to certain lines equals the product of the distances to other lines. See geometry: Cartesian geometry . Fermat emphasized that any relation between x and y coordinates determines a curve (see figure ).

Using this idea, he recast Apollonius’s arguments in algebraic terms and restored lost work. Fermat indicated that any quadratic equation in x and y can be put into the standard form of one of the conic sections. Fermat did not publish his work, and Descartes deliberately made his hard to read in order to discourage “dabblers.” Their ideas gained general acceptance only through the efforts of other mathematicians in the latter half of the 17th century. In particular, the Dutch mathematician Frans van Schooten translated Descartes’s writings from French to Latin. He added vital explanatory material, as did the French lawyer Florimond de Beaune, and the Dutch mathematician Johan de Witt. In England, the mathematician John Wallis popularized analytic geometry, using equations to define conics and derive their properties. He used negative coordinates freely, although it was Isaac Newton who unequivocally used two (oblique) axes to divide the plane into four quadrants, as shown in the figure . Britannica Stories Scientists Ponder Menopause in Killer Whales Analytic geometry had its greatest impact on mathematics via calculus .

Without access to the power of analytic geometry, classical Greek mathematicians such as Archimedes (c. 285–212/211 bc) solved special cases of the basic problems of calculus: finding tangents and extreme points (differential calculus) and arc lengths, areas, and volumes (integral calculus). Renaissance mathematicians were led back to these problems by the needs of astronomy, optics, navigation, warfare, and commerce. They naturally sought to use the power of algebra to define and analyze a growing range of curves. Numbers and Mathematics Fermat developed an algebraic algorithm for finding the tangent to an algebraic curve at a point by finding a line that has a double intersection with the curve at the point—in essence, inventing differential calculus. Descartes introduced a similar but more complicated algorithm using a circle . Fermat computed areas under the curves y = axk for all rational numbers k ≠ −1 by summing areas of inscribed and circumscribed rectangles.

(See exhaustion, method of .) For the rest of the 17th century, the groundwork for calculus was continued by many mathematicians, including the Frenchman Gilles Personne de Roberval , the Italian Bonaventura Cavalieri , and the Britons James Gregory , John Wallis, and Isaac Barrow . Newton and the German Gottfried Leibniz revolutionized mathematics at the end of the 17th century by independently demonstrating the power of calculus. Both men used coordinates to develop notations that expressed the ideas of calculus in full generality and led naturally to differentiation rules and the fundamental theorem of calculus (connecting differential and integral calculus). See analysis . Newton demonstrated the importance of analytic methods in geometry, apart from their role in calculus, when he asserted that any cubic—or, algebraic curve of degree three—has one of four standard equations,xy2 + ey = ax3 + bx2 + cx + d,xy = ax3 + bx2 + cx + d,y2 = ax3 + bx2 + cx + d,y = ax3 + bx2 + cx + d, for suitable coordinate axes. The Scottish mathematician James Stirling proved this assertion in 1717, possibly with Newton’s aid.

Newton divided cubics into 72 species, a total later corrected to 78. Newton also showed how to express an algebraic curve near the origin in terms of the fractional power series y = a1x1/k + a2x2/k + … for a positive integer k. Mathematicians have since used this technique to study algebraic curves of all degrees. Analytic geometry of three and more dimensions Connect with Britannica Facebook Twitter YouTube Instagram Pinterest Although both Descartes and Fermat suggested using three coordinates to study curves and surfaces in space, three-dimensional analytic geometry developed slowly until about 1730, when the Swiss mathematicians Leonhard Euler and Jakob Hermann and the French mathematician Alexis Clairaut produced general equations for cylinders, cones, and surfaces of revolution. For example, Euler and Hermann showed that the equation f(z) = x2 + y2 gives the surface that is produced by revolving the curve f(z) = x2 about the z-axis (see the figure , which shows the elliptic paraboloid z = x2 + y2). Newton made the remarkable claim that all plane cubics arise from those in his third standard form by projection between planes.

This was proved independently in 1731 by Clairaut and the French mathematician François Nicole. Clairaut obtained all the cubics in Newton’s four standard forms as sections of the cubical conezy2 = ax3 + bx2z + cxz2 + dz3 consisting of the lines in space that join the origin (0, 0, 0) to the points on the third standard cubic in the plane z = 1. In 1748 Euler used equations for rotations and translations in space to transform the general quadric surfaceax2 + by2 + cz2 + dxy + exz + fyz + gx + hy + iz + j = 0 so that its principal axes coincide with the coordinate axes. Euler and the French mathematicians Joseph-Louis Lagrange and Gaspard Monge made analytic geometry independent of synthetic (nonanalytic) geometry. Vector analysis In Euclidean space of any dimension , vectors —directed line segments—can be specified by coordinates. An n-tuple (a1, …, an) represents the vector in n-dimensional space that projects onto the real numbers a1, …, an on the coordinate axes.

In 1843 the Irish mathematician-astronomer William Rowan Hamilton represented four-dimensional vectors algebraically and invented the quaternions , the first noncommutative algebra to be extensively studied. Multiplying quaternions with one coordinate zero led Hamilton to discover fundamental operations on vectors. Nevertheless, mathematical physicists found the notation used in vector analysis more flexible—in particular, it is readily extendable to infinite-dimensional spaces. The quaternions remained of interest algebraically and were incorporated in the 1960s into certain new particle physics models. Projections A Biography of Rene Descartes: Creator of the Cartesian Coordinate Plane Birthplace: La Haye en Touraine, Indre-et-Loire, France He was born in France, but spent most of his adult life in the Dutch Republic. His philosophical writing is influential to many, and some refer to him as the Father or Modern Philosophy. He continued his work up through his final days, which he spent in Sweden. Descartes was well known and well respected amongst his peers. Mathematical Legacy Descartes is credited as the first person to use superscripts to represent exponents.

In otherwords, he wrote x•x•x•x = x4 before anyone else. More >> Additioanlly Descartes is credited with providing the groundwork for what Sir Isaac Newton and Gottfried Leibniz independently used to help create modern day calculus. Specifically he used calculus principles for a line of tangent - which is a line that intersects a circle in exactly one point. He also contributed to the field of optics, by identifying laws of reflection and refraction, and identifying that a rainbow's radius is 42°. However, Descartes is probably most famous for his work in analytical geometry. The coordinate plane is commonly referred to as the Cartesian Coordinate because Descartes was so instrumental in developing it. Essentially Descartes was the first person to use algebra to describe geometry. So if you love solving algebra problems in your geometry class, you have Descartes to thank (we won't talk about anyone who doesn't love algebra)! Because of his work, Descartes is commonly refered to as the father or analytic geometry. What books did Rene Descartes write? | Reference.com What books did Rene Descartes write?

A: Quick Answer Descartes' works include "Musicae Compendium," "The World," "Man," "Dioptrics," "The Meteors," "Geometry," "Discourse on the Method," "The Meditations," "Principles of Philosophy," "The Search for Truth," "The Description of the Human Body" and "Passions of the Soul." Some were published posthumously as were some essays and letters that were lesser known. Full Answer Descartes was a French scholar, mathematician and philosopher who lived from 1596 to 1650. He is credited as the father of modern Western philosophy, analytic geometry and the Cartesian coordinate system (named for the Latin translation of Descartes' name). He also played a key role in the scientific revolution of Europe that subsequently lead to the social movement of the Enlightenment era. The quote "I think therefore I am," remains the hallmark of Descartes' philosophy. It appeared for the first time in "Discourse on the Method," but was also referenced in later works.

Descartes posited that because man can err in reasoning or be mislead by his senses, he must strip away all he believed was real and view it as merely an illusion. Inferring, however, that there is a being, a man who is capable of perceiving or pondering himself and his environment at all was undeniable proof that he, at the very least, existed. This proof became a pillar for many of Descartes' later arguments. Rene Descartes - Father of Modern Philosophy - on Mathematics - YouTube Rene Descartes - Father of Modern Philosophy - on Mathematics Want to watch this again later? Sign in to add this video to a playlist. Need to report the video? Sign in to report inappropriate content. The interactive transcript could not be loaded. Loading... Rating is available when the video has been rented. This feature is not available right now. Please try again later. Uploaded on Sep 1, 2011 René Descartes (31 March 1596 -- 11 February 1650) was a French philosopher and writer who spent most of his adult life in the Dutch Republic.

He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day. In particular, his Meditations on First Philosophy continues to be a standard text at most university philosophy departments. Descartes' influence in mathematics is equally apparent; the Cartesian coordinate system—allowing algebraic equations to be expressed as geometric shapes (2D coordinate system)—was named after him. He is credited as the father of analytical geometry, the bridge between algebra and geometry, crucial to the discovery of infinitesimal calculus and analysis. Descartes was also one of the key figures in the Scientific Revolution. Category TheScienceClassroom - René Descartes René Descartes Export (PDF) René Descartes Rene Descartes mainly spent most of his lifetime researching in the Dutch Republic. There he made many discoveries and led to the advancement of two major subject in the world. Those major subjects were mathematics and philosophy. Widely known as many things, he was respectively known as a French philosopher and physicist. Descartes was called " the father of analytical geometry". He took the world of math to another level. He invented the Cartesian coordinate system.

He connected the bridge between algebra and geometry. Descartes not only widely influencing the world of mathematics, he payed a great deal of this lifetime to the world of philosophy.Descartes was categorized in natural philosophy for his own philosophical theories, Mind-body Dualism .[ ⁶ ] He had his own separate ideas, beliefs, and thoughts. Therefore in history, he was known as his own thinker. Descartes' most important theology was the existence of God. He believed God had absolute freedom in the world's creation and its living things. Using his newly profound method of doubt, he cleared up any accusations from the two major philosophy groups of his era, Aristotelian ism and the Schools, that related to God's existence. For that reason, Descartes was heavily looked down upon by the empiricist schools of thoughts, which consisted of Hobbes, Locke, Berkeley And Hume. Today, Descartes is recognized as " the father of modern day philosophy". He influential changed the way philosophers thought and felt about ideas. For that reason, philosophy today owes him a lifetime of gratitude.

One of his remember able phrase " Je pense, donc je suis" meaning "I think, therefore I am; or I am thinking, therefore I exist" still lives in the minds of many great scientists,mathematicians, and philosophers today. Respected by all and hated by few, Rene Descartes greatly contributed to the world of mathematics and philosophy.[ ⁴ ][ ⁵ ] Insight and Influences Image Courtesy of thesplendorofthechurch Rene Descartes was born and raised in La Haye, France: he was integrated into a high class family as a successor. Descartes' parents were Joachim Descartes and Jeanne Brochard. His father was a lawyer and magistrate. For that reason, Descartes was acknowledged to follow his father's footsteps. One year after giving birth to Descartes, Jeanee Brochard, his mother, died of tuberculosis. Since Joachim Descartes was a lawyer, he had very little time dedicated to his kids. Therefore, young Descartes , his full sister and brother, Jeanne and Pierre, were entrusted to their grandmother in order to be taken care of.

In 1607, at the age of ten, young Descartes was sent to the Jesuit college of La Fleche. There he studied the required courses, grammar, philosophy, and mathematics. After completing his studies at the Jesuit college of La Fleche in 1614, in order to please his father, he wanted to further his knowledge in law. In the month of May, a year after his completion at Jesuit college, he attended the University of Poitier. At the university, in 1616, he acquired his Baccalaureate and License in Canon & Civil Law. Descartes was living up to his father's expectations.[ ¹ ][ ³ ][ ⁵ ] Descartes was not born into the world physics. It was later in life that he was influenced by a well recognized man. It was then that his interest with physics began. It first started when Descartes was at a fine age of twenty-two. The year was 1618. Descartes had enlisted in the military army of the Prince Maurice of Nassa. His intentions were to travel the world in search for a "truth" in life.

Clearly defined in his later book, Discourse on the Method of Rightly Conducting One's Reason and Seeking the Truth in the Sciences , of his yearned for "truth" in his researching, "I entirely abandoned the study of letters. Resolving to seek no knowledge other than that which could be found in myself or else in the great book of the world, I spent the rest of my youth traveling, visiting courts and armies, mixing with people of diverse temperaments and ranks, gathering various experiences, testing myself in the situations which fortune offered me, and at all times reflecting upon whatever came my way so as to derive some profit from it."[ ² ][ ³ ][ ⁴ ] While traveling with the army, Descartes met a well recognized physicist/scientist by the name of Isaac Beeckman. Beeckman was the reason behind Descartes’ fame in the world of mathematics. He was Descartes greatest childhood influence. He sparked Descartes first interest in the world of physics. He challenged Descartes by presenting mathematics applied to other fields. Fascinated by Beeckman’s knowledge, Descartes began studying under Beeckman’s guidance.

Studying under Beeckman, Descartes took serious interest in mechanical problems and mathematics. Beeckman founded the answer to Descartes’ desire for the “truth” by the use of mathematics. In 1621, Descartes resigned from the army, and continued his travels independently. Settled in Paris, in 1626, Descartes lived for a period of his life awaiting for the theory of the universe to come to him. His full attention was towards the study of mathematics, science, and philosophy.[ ⁴ ] From then on, the dreams his father once had of him entering Parliament was long gone. Descartes had followed his own separate way, the path of true wisdom of the diverse countries. Major Contributions Rene Descartes, not only contributing to one subject, he contributed to all the major subjects in the world. He gave mathematics a new method, "analytical geometry". He connected missing links and answered questionable methods in mathematics. He also played a vital role in the invention of calculus. Therefore, mathematicians, with the aid of Descartes, cleared another road block in order for someday to finally comprehend the mysteries of algebra, geometry, and mathematics as a whole.

In the world of philosophy, Descartes set a high standard in the eyes of philosophers today. He is considered as one of the most influential thinkers of his time. Today, he is declared as "The Father of Modern Philosophy". He believed in order to find "truth", one would have to become unbiased in situation. In his book Meditations, he says "If you would be a real seeker after truth, it is necessary that at least once in your life you doubt, as far as possible, all things."[ ⁹ ] Analytical Geometry, created by Descartes, is the link which connected algebra and geometry. Descartes' inspiration, Isaac Beeckman, was the missing key to unlock Descartes' interest in math and science. Beeckman and Descartes challenged each others ideas and beliefs. While also questioning each others theories, they both unconditionally respected each others ideas and beliefs. In 1618, while suggesting to Beeckman a more efficient way to solve his theory, Descartes applied algebraic formulas to solve geometric problems in order complete the uncompleted equation.

In that same year, a section in Beeckman's journal described Descartes remarkable method of using the foundation of algebra and applying it to geometry. For that reason, Descartes was titled "the father of analytical geometry". Descartes stated every subject, mainly directed at science, used math in order to solve daily life situations, writing philosophy, and defying questionable theories of science. Therefore, he believed math and science are intertwined together and could not live without one or the other. In his eyes, math was science , and science was math. They were the same subject in his mind.[ ⁹ ] As for philosophy, Descartes gave a great ordeal of his lifetime questioning the old century philosophy. During his time period, Scholastic-Aristotelian philosophy played an influenced on Descartes' ideas; Scholastic-Aristotelian philosophy was considered the high of the philosophical world. Descartes strongly supported the accusations of doubt. He referenced "doubt" as a false in the mind. He was one of the very first man that challenged skepticism.[ ¹⁰ ] He stated in order not to limit oneself of knowledge, one must doubt.

Quoted by Rene Descartes, "The long concatenations of simple and easy reasoning which geometricians use in achieving their most difficult demonstrations gave me occasion to imagine that all matters which may enter the human mind were interrelated in the same fashion.", clearly supports his method of doubt.[ ⁹ ] Therefore, credited to Descartes, he created modern philosophy and what it stands for today. Descartes' philosophy questioned the church's existence. In that period, any questions relating to the church would be shunned or demolished by the church itself. If an event happened to disrupt the peace of the church, tragic deeds took place, which would rapidly obliterate all roots leading to the main source. For that reason, Descartes stayed in secrecy. Affect and Effect Image Courtesy of suite101 A gifted child at birth, Descartes, was easily fascinated, but not persuadable. Rarely relying on experiences, Descartes states doubt blinds one's complete knowledge of life itself. Hence, he neglected the wisdom attained from knowledge by past experiences, Empiricism. Descartes was a strong and passionate man of knowledge by reasons and hard facts, Rationalism.

In his book, Meditations, he respectively defines why reason and facts clears the darkened passage towards true knowledge. In one of his quote, he claims Common sense is the most fairly distributed thing in the world, for each one thinks he is so well-endowed with it that even those who are hardest to satisfy in all other matters are not in the habit of desiring more of it than they already have."[ ¹⁵ ] Strongly opinionated, Descartes had his own separate mindset and beliefs. Descartes solving method differed from other philosophical greats.His method was unbiased and neutral. He believed that was the only right way of acquiring true knowledge from life. Descartes believed reasoning and doubt was behind the questions of life, excluding the existence of God. A man of certainty, Descartes was a Catholic. He acknowledges the existence of God. Therefore, he states philosophical judgment does not intertwine with religion. Even towards the end of his days, Descartes intentionally avoided any situations involving God's existence. In his mind, God existed in everything and was independent of any forces, beings, and theories. As a result, God's existence cannot be questioned or doubted.

Therefore Referencing back to his quote, "I think, therefore, I am," and "I think, therefore there is a God!"[ ¹⁴ ], clearly proves his passionate belief in God's existence. Without thought, he acclaims, humans and life cannot exist. Hence, there is a God.[ ¹⁴ ] For that reason, Descartes' belief in the existence of God played one of the very many major influences in his life, and possibly could have limited his path towards true knowledge of the world. Highly Influenced by a divine being, Rene Descartes also had many other influences. One major vital role to his life was not an uncountable divine being, but a influential human being. Descartes praised and highly respected this human being. This person was Sir Isaac Beeckman, a well known Dutch philosopher and scientist. Beeckman challenged Descartes philosophical ideas, and also expressively shared his own ideas. Intrigued by Beeckman, Descartes willingly studied under his influences. Thus, Beeckman was Descartes' teacher ,and Descartes was Beeckman's student.

Therefore, Beeckman not only contributed to Descartes strong-willed and expressive natural philosophical ideas, he is credited to all the life-long works of Descartes.[ ⁴ ][ ¹³ ] Rene Descartes revolutionized modern philosophy. He aided the early days of United States government based ideas and the people within it. He influenced great philosphers like John Locke, an Englishmen credited as one of the greatest influential thinkers of the 17th century in "The Enlightment" era. The Cartesian method, created by Descartes, removed all uncertainties in philosophies. As a result, today in society, people chooses to rely on facts and reasoning, rather than judgment, in order to verify their statements or philosophical ideas.[ ¹³ ][ ¹⁴ ] Legacy Rene Descartes was a well-known French physicist and philospher. During his lifetime, He lead to many discoveries. He answered the questions asked by Image Courtesy of answers many great mathematicians and philosphers of that time period. On Feburary 11, 1650, his life had come to a sudden stop.

Interested in Descartes' knowledge and wisdom, Queen Christiana of Sweden, asked Descartes to come visit her and teach her his philosophy of truth. For that reason, ill Descartes had to wake up at five in the morning; It disrupted his usual sleeping schedule until noon. Hence, It worsened the health of the fragile Descartes, which ended his life of the doubtful world he depicted to live in.[ ¹⁰ ] Descartes' death did not mean the stop of his influences on the world. After his death, he gave the modern thinkers today a pathway to new ideas and discoveries. He also drew a vivid path for mathematicians and scientists. His discovery of "analytical geometry" contributed to all the high class mathematics today. For example, Descartes indirectly benefited the discovery of "calculas" by the later great phycists, Sir Issac Newton and Gottfried Leibeniz, and influenced great thinkers like John Locke. For that reason, Descartes is widely known in both worlds of mathematics and philosphy. In mathematics, Descartes was known as " the father of analytical geometry".

Analytical geometry involved a coordinate plane; coordinate plane called the coordinate plane system was invented also by Descartes. It linked algebra and geometry. Before his time, algebraic and geometric problems were independent subjects. But during Descartes time period, he directly related algebric and geometric problems together. Therefore, he was prized with the honorable declaration as "the father of anayltical geometry".[ ¹⁶ ] Descartes' strong-willed and theology played the vital role to his honorary status. He used many complex equations inorder to answer the complicated questions truth, but yet, he used them so simply and easily. He created four steps. He declared these steps would solve all and any situations presented in life. In his book Discours de la Méthode, he clearly defines his four simple steps that solves and unreveals the complexity of the truth of life. " thought the following four [rules] would be enough, provided that I made a firm and constant resolution not to fail even once in the observance of them.

The first was never to accept anything as true if I had not evident knowledge of its being so; that is, carefully to avoid precipitancy and prejudice, and to embrace in my judgment only what presented itself to my mind so clearly and distinctly that I had no occasion to doubt it. The second, to divide each problem I examined into as many parts as was feasible, and as was requisite for its better solution. The third, to direct my thoughts in an orderly way; beginning with the simplest objects, those most apt to be known, and ascending little by little, in steps as it were, to the knowledge of the most complex; and establishing an order in thought even when the objects had no natural priority one to another. And the last, to make throughout such complete enumerations and such general surveys that I might be sure of leaving nothing out.

These long chains of perfectly simple and easy reasonings by means of which geometers are accustomed to carry out their most difficult demonstrations had led me to fancy that everything that can fall under human knowledge forms a similar sequence; and that so long as we avoid accepting as true what is not so, and always preserve the right order of deduction of one thing from another, there can be nothing too remote to be reached in the end, or to well hidden to be discovered."[ ¹⁶ ] Philosophy & Religion: René Descartes What is Calculus? What is Calculus? While many people believe that calculus is supposed to be a hard math course, most don't have any idea of what it is about. The good news is that if you remember your algebra and are reasonably good at it then calculus is not nearly as difficult as its reputation supposes. This article attempts to explain just what calculus is about--where it came from and why it is important. First, a little history leading up to the discovery of calculus, or its creation, depending on your philosophy. The word "calculus" comes from "rock", and also means a stone formed in a body.

People in ancient times did arithmetic with piles of stones, so a particular method of computation in mathematics came to be known as calculus. Arithmetic and geometry are the two branches of mathematics originating in ancient times. Mathematicians attempted to do algebra in those days but lacked the language of algebra, namely the symbols we take for granted such as +, -, X, ÷ and =. Much of the world, including Europe, also lacked an efficient numbering system such as that developed in the Hindu and Arabic cultures. (Try long division, for example, using Roman numerals.) Algebra as a branch of mathematics can be said to date to around 825 A.D. when a Persian, al-Khwarizmi, wrote the earliest known algebra text. (The word "algebra" comes from a Persian word in the title, "al'jabr", which means "to restore". The English term for a systematic mathematical method, algorithm, was derived from al-Khwarizmi's name by way of a Latin translation.) For over seven hundred years algebra and geometry coexisted but were not well linked. Geometry describes the physical nature of our world while algebra is a sophisticated tool for mathematical analysis.

Due to the Greek influence on Persian (or Islamic) mathematics geometry was successfully used to verify some of their algebraic methods, but there was no known way to harness the analytical power of algebra to analyze geometry. In the late 1500's the French philosopher and mathematician, Rene Descartes, had a profound breakthrough when he realized he could describe position on a plane using a pair of numbers associated with a horizontal axis and a vertical axis. By describing, say, the horizontal measurement with x's and the vertical measurement with y's, Descartes was able to give geometric objects such as lines and circles representation as algebraic equations. This seminal construction of what we call graphs is, arguably, the cornerstone without which our modern technology would not be possible. Descartes thus united the analytical power of algebra with the descriptive power of geometry into a branch of mathematics he called analytic geometry. This term is sometimes seen in textbooks with titles such as "Calculus with Analytic Geometry." Descartes, as philosopher, is also the author of the famous line, "Cogito, ergo sum," or, "I think, therefore I am." He was attempting to settle an argument about whether we exist independently of God's imagination.

The next major breakthrough in mathematics was the discovery (or creation) of calculus around the 1670's. Sir Isaac Newton of England, and a German, Gottfried Wilhelm Leibnitz, deserve equal credit for independently coming up with calculus. Each accused the other of plagiarism for the rest of their lives, but for what it's worth, the world largely adopted Leibnitz's calculus symbols. Calculus did allow Newton to establish physics principles which remained uncontested until the year 1900 and which in our ordinary scale world still suffice to explain physics to excellent accuracy. Calculus was developed out of a need to understand continuously changing quantities. Newton, for example, was trying to understand the effect of gravity which causes falling objects to constantly accelerate. The speed of an object increases constantly every split second as it falls. How can one, for example, determine the speed of a falling object at a frozen instant in time, such as its speed when it strikes the ground? No mathematics prior to Newton and Leibnitz's time could answer such a question, which appeared to amount to the impossibility of dividing zero by zero. The solution to this type of issue came to be known as the derivative.

Derivatives are slopes of particular lines called tangent lines, and the reader may recall that slope of a line is a concept from Descartes' graphing. Differential calculus is one side of calculus, the part concerned with continuous change and its applications. By understanding derivatives the student has at his or her disposal a very powerful tool for understanding the behavior of mathematical functions. Importantly, this allows us to optimize functions, which means to find their maximum or minimum values, as well as to determine other valuable qualities describing functions. Real-world applications are endless, but some examples are maximizing profit, minimizing stress, maximizing efficiency, minimizing cost, finding the point of diminishing returns, and determining velocity and acceleration. The other primary side of calculus is integral calculus. Integration is a process which, simplistically, resembles the reverse of differentiation. This amounts to efficiently adding infinitely many infinitely small numbers. This allows us, in theory, to find the area of any planar geometric shape, or the volume of any geometric solid. But the applications of integration, like differentiation, are also quite extensive. Until the mid-1800's mathematicians were content to use calculus-style computations under the heuristic evidence that they seemed to work very well.

This was a fragile house of cards increasingly based on the faith that what they saw would always work. Largely under the influence of Karl Friederich Gauss (1777 - 1855) the mathematical world gradually returned to the ancient Greek ideal of mathematical proof by logic found in their [Euclidean] geometry. Gauss' student, Bernhard Riemann (1826 - 1866), and some of his contemporaries established a rigorous logical foundation for calculus now known as real analysis. Their definitions and theorems greatly influenced the language and teaching of calculus today. It was only through calculus and the rigorous treatment it received in the 19th century that mankind could really begin to grasp the difficult concepts of infinity and infinitesimal. Calculus also completes the link of algebra and geometry by providing powerful analytical tools that allow us to understand algebra functions through their related geometry. We now realize that great thinkers in ancient times ran into calculus concepts. Archimedes used calculus thinking, for example, to establish the area of a circle and the volume of a sphere, borrowing his methods of exhaustion--essentially limits--from Eudoxus of Cnidus.

Zeno of Elea proposed four famous paradoxes which caused Aristotle, centuries later, to grapple with calculus ideas in his failed attempt to resolve them. Calculus, by tradition, is usually a one-year course (four quarters or three semesters). The first half is concerned with learning and applying the techniques of differentiation and integration. The second half is concerned with further applications, using both sides of calculus, to vectors, infinite sums, differential equations and a few other topics. The last term of calculus is sometimes known as multivariate calculus, which is an application of calculus to three or more dimensions. Calculus provides the foundation to physics, engineering, and many higher math courses. It is also important to chemistry, astronomy, economics and statistics. Medical schools and pharmacy schools use it as a screening tool to weed out weaker aspirants under the assumption that people who are unwilling or unable to handle the rigors of calculus stand little chance of surviving the hard work of studying medicine or pharmacology. There are three main facets to being a successful calculus student: --You must be good at algebra skills. It is not enough to have passed algebra, you must also remember what you learned! If you have to relearn algebra while learning calculus then the burden can overwhelm.

--Memorization of computational patterns is not enough. Some people can get by in algebra by memorization without understanding. In calculus it is quite necessary to pay attention and learn the concepts in order to apply them. This is learning at a mature level. --You must be dedicated to study. Don't skip any classes except for the most dire reasons. Take notes. Above all, practice lots of problems, without which those concepts will not be reinforced and learned. Students who enjoy intellectual stimulation and the power of abstract thinking tend to enjoy the beauty of calculus the most, but there is much to appreciate for those who are looking for powerful tools which which to understand and create in the physical world. Finally, a good reason to take calculus is that you will be more competitive and have more career opportunities. Many people avoid demanding challenges; those willing to face them head on tend to go much further in life.

To contact the author by e-mail click on this link: Jon Davidson 17th Century Mathematics - The Story of Mathematics The Story of Mathematics 17TH CENTURY MATHEMATICS Logarithms were invented by John Napier, early in the 17th Century In the wake of the Renaissance, the 17th Century saw an unprecedented explosion of mathematical and scientific ideas across Europe, a period sometimes called the Age of Reason. Hard on the heels of the �Copernican Revolution� of Nicolaus Copernicus in the 16th Century, scientists like Galileo Galilei, Tycho Brahe and Johannes Kepler were making equally revolutionary discoveries in the exploration of the Solar system, leading to Kepler�s formulation of mathematical laws of planetary motion. The invention of the logarithm in the early 17th Century by John Napier (and later improved by Napier and Henry Briggs) contributed to the advance of science, astronomy and mathematics by making some difficult calculations relatively easy. It was one of the most significant mathematical developments of the age, and 17th Century physicists like Kepler and Newton could never have performed the complex calculatons needed for their innovations without it.

The French astronomer and mathematician Pierre Simon Laplace remarked, almost two centuries later, that Napier, by halving the labours of astronomers, had doubled their lifetimes. The logarithm of a number is the exponent when that number is expressed as a power of 10 (or any other base). It is effectively the inverse of exponentiation. For example, the base 10 logarithm of 100 (usually written log10 100 or lg 100 or just log 100) is 2, because 102 = 100. The value of logarithms arises from the fact that multiplication of two or more numbers is equivalent to adding their logarithms, a much simpler operation. In the same way, division involves the subtraction of logarithms, squaring is as simple as multiplying the logarithm by two (or by three for cubing, etc), square roots requires dividing the logarithm by 2 (or by 3 for cube roots, etc). Although base 10 is the most popular base, another common base for logarithms is the number e which has a value of 2.7182818...

and which has special properties which make it very useful for logarithmic calculations. These are known as natural logarithms, and are written loge or ln. Briggs produced extensive lookup tables of common (base 10) logarithms, and by 1622 William Oughted had produced a logarithmic slide rule, an instrument which became indispensible in technological innovation for the next 300 years. Napier also improved Simon Stevin's decimal notation and popularized the use of the decimal point, and made lattice multiplication (originally developed by the Persian mathematician Al-Khwarizmi and introduced into Europe by Fibonacci ) more convenient with the introduction of �Napier's Bones�, a multiplication tool using a set of numbered rods. Graph of the number of digits in the known Mersenne primes Although not principally a mathematician, the role of the Frenchman Marin Mersenne as a sort of clearing house and go-between for mathematical thought in France during this period was crucial. Mersenne is largely remembered in mathematics today in the term Mersenne primes - prime numbers that are one less than a power of 2, e.g.

3 (22-1), 7 (23-1), 31 (25-1), 127 (27-1), 8191 (213-1), etc. In modern times, the largest known prime number has almost always been a Mersenne prime, but in actual fact, Mersenne�s real connection with the numbers was only to compile a none-too-accurate list of the smaller ones (when Edouard Lucas devised a method of checking them in the 19th Century, he pointed out that Mersenne had incorrectly included 267-1 and left out 261-1, 289-1 and 2107-1 from his list). The Frenchman Ren� Descartes is sometimes considered the first of the modern school of mathematics. His development of analytic geometry and Cartesian coordinates in the mid-17th Century soon allowed the orbits of the planets to be plotted on a graph, as well as laying the foundations for the later development of calculus (and much later multi-dimensional geometry). Descartes is also credited with the first use of superscripts for powers or exponents.

Two other great French mathematicians were close contemporaries of Descartes : Pierre de Fermat and Blaise Pascal . Fermat formulated several theorems which greatly extended our knowlege of number theory, as well as contributing some early work on infinitesimal calculus. Pascal is most famous for Pascal�s Triangle of binomial coefficients, although similar figures had actually been produced by Chinese and Persian mathematicians long before him. It was an ongoing exchange of letters between Fermat and Pascal that led to the development of the concept of expected values and the field of probability theory. The first published work on probability theory, however, and the first to outline the concept of mathematical expectation, was by the Dutchman Christiaan Huygens in 1657, although it was largely based on the ideas in the letters of the two Frenchmen. Desargues� perspective theorem The French mathematician and engineer Girard Desargues is considered one of the founders of the field of projective geometry, later developed further by Jean Victor Poncelet and Gaspard Monge. Projective geometry considers what happens to shapes when they are projected on to a non-parallel plane.

For example, a circle may be projected into an ellipse or a hyperbola, and so these curves may all be regarded as equivalent in projective geometry. In particular, Desargues developed the pivotal concept of the �point at infinity� where parallels actually meet. His perspective theorem states that, when two triangles are in perspective, their corresponding sides meet at points on the same collinear line. By �standing on the shoulders of giants�, the Englishman Sir Isaac Newton was able to pin down the laws of physics in an unprecedented way, and he effectively laid the groundwork for all of classical mechanics, almost single-handedly. But his contribution to mathematics should never be underestimated, and nowadays he is often considered, along with Archimedes and Gauss , as one of the greatest mathematicians of all time. Newton and, independently, the German philosopher and mathematician Gottfried Leibniz , completely revolutionized mathematics (not to mention physics, engineering, economics and science in general) by the development of infinitesimal calculus, with its two main operations, differentiation and integration. Newton probably developed his work before Leibniz , but Leibniz published his first, leading to an extended and rancorous dispute.

Whatever the truth behind the various claims, though, it is Leibniz �s calculus notation that is the one still in use today, and calculus of some sort is used extensively in everything from engineering to economics to medicine to astronomy. Both Newton and Leibniz also contributed greatly in other areas of mathematics, including Newton �s contributions to a generalized binomial theorem, the theory of finite differences and the use of infinite power series, and Leibniz �s development of a mechanical forerunner to the computer and the use of matrices to solve linear equations. However, credit should also be given to some earlier 17th Century mathematicians whose work partially anticipated, and to some extent paved the way for, the development of infinitesimal calculus. As early as the 1630s, the Italian mathematician Bonaventura Cavalieri developed a geometrical approach to calculus known as Cavalieri's principle, or the �method of indivisibles�.

The Englishman John Wallis, who systematized and extended the methods of analysis of Descartes and Cavalieri, also made significant contributions towards the development of calculus, as well as originating the idea of the number line, introducing the symbol ∞ for infinity and the term �continued fraction�, and extending the standard notation for powers to include negative integers and rational numbers. Newton 's teacher Isaac Barrow is usually credited with the discovery (or at least the first rigorous statrement of) the fundamental theorem of calculus, which essentially showed that integration and differentiation are inverse operations, and he also made complete translations of Euclid into Latin and English.

Question: Which French philosospher is credited with creating analytical geometry?
Answer: DESCARTES